Queen the bastard hot tub would have been bastard expensive, so you are well rid.
Queen and Lonely - Green army! ✊🏻
I like to see Fluffy hitting her stride. Fluffy, I have visions of you rolling your sleeves up and cracking your knuckles ready to help Queen 
Sunny it's a big question and I don't know whether my views are at all worthwhile as I'm not an expert and this is a very personal dilemma . But here goes:
I think a diagnosis would (should?) only ever really be an advantage. My understanding is that the diagnostic threshold is meant to be met if the person's ASD traits have an impact on their day-to-day life. And if that is the case the ability to use the label to explain those difficulties and access reasonable adjustments and support services should be helpful. And as Cag says, it's up to you when and whether you choose to tell people. If its not going to be helpful then you don't. I think Need's post about her DN's experience was really helpful (and encouraging to me too so thanks Need !) . I think it's also encouraging that both Cag and Meadow have found their diagnoses helpful. I hope my DS will also find it helpful to understand that there are other people like him, and to understand why he finds some things harder/easier than most people.
I completely understand your concern that you don't want anyone to lower their aspirations for your DS. I hope that wouldn't happen - it's early days for us but my experience is that people are just as likely to expect a great deal academically because of the label (not necessarily justified - but there is a perception that if he is autistic DS must be a quirky genius 
). It is a very broad diagnosis indeed and I think in education they generally understand that. While they may expect social difficulties etc I think they are genuinely open minded about academic potential.
Re. Private school I don't think they could refuse him entry on the basis of a diagnosis but I suppose they could if he needed more support than they were able to offer, ie beyond 'reasonable adjustments' . One of my friends is a teacher in a posh public school and he teaches a number of HFA boys. It's no barrier where he is. TBH, as someone said upthread (Cag?) if this particular school latches onto an ASD diagnosis as a way of wriggling out of taking your DS then chances are it wouldn't be a great place to be. A genuinely inclusive attitude is as important as academic offerings, IMO.
But all that said only you can know whether it feels right for you all. Very good luck with grappling with your dilemma and I hope it works out well for your DS.
I have been typing very slowly while watching Strictly on catch up. I agree about Susan's quick step Cag - lovely! I am sure I've cross posted with loads of people...